Let $\{n_k, k \geqq 1\}$ be a sequence of random variables uniformly distributed over $\{0, 1\}$ and let $F_N(t)$ be the empirical distribution function at stage $N$. Put $f_n(t) = N(F_N(t) - t)(N\log\log N)^{-\frac{1}{2}}, 0 \leqq t \leqq 1, N \geqq 3$. For strictly stationary sequences $\{n_k\}$ where $n_k$ is a function of random variables satisfying a strong mixing condition or where $n_k = n_k x \mod 1$ with $\{n_k, k \geqq 1\}$ a lacunary sequence of real numbers a functional law of the iterated longarithm is proven: The sequence $\{f_N(t), N \geqq 3\}$ is with probability 1 relatively compact in $D\lbrack 0, 1\rbrack$ and the set of its limits is the unit ball in the reproducing kernel Hilbert space associated with the covariance function of the appropriate Gaussian process.

Let $\{X(t): t\in\lbrack 0, 1\rbrack\}$ be a stochastic process. For any function $f$ such that $E(X(t) - X(s))^2 \leqq f(|t - s|)$, a condition is found which implies that $X$ is sample-continuous and satisfies the central limit theorem in $C\lbrack 0, 1\rbrack$. Counterexamples are constructed to verify a conjecture of Garsia and Rodemich and to improve a result of Dudley.

Let $\{X(t): t\in\lbrack 0, 1\rbrack\}$ be a stochastic process and $f$ a nonnegative function on [0, 1] which is nondecreasing in a neighborhood of 0. Under the assumption that $E(X(t) - X(s))^2 \leqq f(|t - s|)$, we find best possible conditions for determining whether or not $X(t)$ is sample-continuous.

Call a Markov process "ergodic" if the following conditions hold: (a) The process has a unique invariant measure $\nu$. (b) If $\mu_0$ is any initial distribution for the process, then the resulting distribution $\mu_t$ at time $t$ will converge weakly to $\nu$ as $t \rightarrow \infty$. In this paper, necessary and sufficient conditions are obtained for the ergodicity of a certain infinite particle process. This process models a dissonant voting system, and is similar to one treated in Holley and Liggett (1975).

Let $\mathcal{N}$ denote the collection of all Radon measures $n$ on $\mathbb{R}$ such that $0 < \lim_{x\rightarrow\infty} n((0, x\rbrack)/x = \lim_{x\rightarrow -\infty}n((x, 0\rbrack)/|x| < \infty$. For $n\in\mathscr{N}$, let $n^{-1}\in\mathscr{N}$ be the measure whose distribution function is the inverse of the distribution function of $n$. Given a random element $N$ of $\mathscr{N}$ having distribution $P$, let $P^I$ denote the distribution of $N^{-1}$. Let $N$ be a random element of $\mathscr{N}$ having stationary distribution $P$ and let $P^T$ be the appropriately defined tagged distribution corresponding to $P$. It is shown that $P^I$ has an asymptotically stationary distribution $P^S$ on $\mathscr{N}$. Moreover $P = (P^S)^S, P^I = (P^S)^T$, and $P^T = (P^S)^I. P^S$ is given explicitly in terms of $P^T$. In particular, if $N$ is purely nonatomic with probability one, then $P^S = (P^T)^I$. If $P$ is a stationary compound renewal process, then so is $P^S$.

A limit theorem is established for the length of the longest chain of random values in $R^d$ with respect to a partial ordering. The result is applied to a question raised by T. Robertson and F. T. Wright concerning the generalized empirical distribution function associated with the class of lower layers.

We prove that the probability of at least $k$ successes, in $n$ Bernoulli trials with success-probability $p$, is larger than its normal approximant if $p \leqq \frac{1}{4}$ and $k \geqq np$ or if $p \leqq \frac{1}{2}$ and $np \leqq k \leqq n(1 - p)$. A local refinement is given for $np \leqq k \leqq n(1 - p), k \geqq 2$, and for $p \leqq \frac{1}{4}, k \geqq n(1 - p)$. Bounds below for individual binomial probabilities $b(k, n, p)$ are also given under various conditions. Finally, we discuss applications to significance tests in one-way layouts.

Consider a system subject to periods of deterioration. The system might fail at any time within the set of deterioration time, and the probability of failure is a function of the accumulated damage caused from past deterioration. When the system fails, it is immediately replaced and a failure cost is incurred; if replacement is made before failure, a lesser cost is incurred and that cost may depend upon the amount of accumulated damage at the replacement time. The purpose of this paper is to derive the optimal replacement policy for such a system whose set of deterioration times contains no isolated points and whose cumulative damage process is a semi-Markov process. Only those policies which make a replacement within the set of deterioration times are considered. Optimality is based on a discounted cost criterion.

Given a Markov chain with stationary transition probabilities, we study random times $\tau$ determined by the evolution of the Markov chain for which either the pre-$\tau$ or post-$\tau$ process is Markovian with stationary transition probabilities. A complete description is given of all such random times which admit a conditional independence property analogous to the strong Markov property at a stopping time.

Let $E$ be a finite partially ordered set and $M_p$ the set of probability measures in $E$ giving a positive correlation to each pair of increasing functions on $E$. Given a Markov process with state space $E$ whose transition operator (on functions) maps increasing functions into increasing functions, let $U_t$ be the transition operator on measures. In order that $U_tM_p \subset M_p$ for each $t \geqq 0$, it is necessary and sufficient that every jump of the sample paths is up or down.

For stable processes of index $\alpha, 1 < \alpha < 2$, exact upper functions are determined for the sample function growth at a last exit time.

Let $Z_0(t)$ be a Markov chain and $X(t)$ a subordinator. Set $Z(t) = Z_0(X(t))$ and let $\alpha$ be a nonstable state of $Z_0$. It is shown, via an example, that it is possible for $\alpha$ to be a stable state of $Z(t)$ even when the total mass of the Levy measure of $X$ is unbounded.

This paper constructs a collection of probability vectors $\varphi_n$ for all $n\in\mathbf{Z}$ and a stochastic matrix $Q$ on a countable state space so that (1) $$Q(i, j) > 0 \text{for all} i, j$$, (2) $$\varphi_nQ = \varphi_{n+1} \text{for all} n\in\mathbf{Z}$$, (3) $$\varphi_n = \varphi_{n+1} \text{for all} n \geqq 0; _{\varphi-1} \neq \varphi_0$$.

If $n(x)$ is the standard normal density on $R^2$ and if $A = -A$ and $B = -B$ are convex subsets of $R^2$ then $$\int_{A\cap B}\mathbf{n}(x) d^2x \geqq (\int_A \mathbf{n}(x) d^2x)(\int_B \mathbf{n}(x) d^2x).$$

It is shown that every nonnegative superfair process (in particular a nonnegative submartingale) is the absolute value of a symmetric fair process (martingale). Is every submartingale a convex function of a martingale? No. If however the adjective convex is omitted from the question, an affirmative answer is provided. Furthermore, transforming functions $\phi$, such that every superfair process (submartingale) is that $\phi$ of a fair process (martingale), are shown to exist. The results are extended to continuous-parameter submartingales with rightcontinuous sample functions.

A coupling method is used to give another proof of Blackwell's renewal theorem. The nature of the proof is probabilistic, using in an essential way the nonlattice property of the lifetime distribution and the Hewitt-Savage zero-one law.

Assume a population is distributed in an infinite lattice of colonies in a migration and random-mating model in which all creatures are selectively equivalent. In one and two dimensions, the population tends to consolidate into larger and larger blocks, each of which is composed of the descendents of a single initial individual. The purpose here is to describe the variation of the size and shape of these blocks with time. Specifically, we obtain asymptotic results for (1) the expected number of individuals in, (2) the approximate radius of, and (3) the distribution of the individuals within a typical block for large time. These results depend on the dimension, and most extend to three or more dimensions.

It is proved that if $\mu$ is a stable measure of index $p$ on a vector space and $q$ is a seminorm, then $\lim_{t\rightarrow\infty} t^p\mu\{x: q(x) > t\}$ exists.

Under the condition that the minimal sufficient statistics are transitive, the sequence of Rao-Blackwell estimators of distribution function has been shown to form a reverse martingale sequence. Weak convergence of the corresponding empirical process to a Gaussian process has been established by assuming that the sufficient statistics are $U$-statistics and utilizing certain results on the convergence of conditional expectations of functions of $U$-statistics along with the functional central limit theorems for (reverse) martingales by Loynes (1970) and Brown (1971).